Databases Reference
In-Depth Information
Example 3.
Consider a set
B
for persons of medium height. Based on common
sense, it is partially true that a 1.7 m person is of medium height with a degree
of 0
.
7. Also, consider a union set
E
of tall and medium height people, and a
union set
F
of medium height and short people. We can believe that a 1.7 m
person belongs to
E
and
F
with an increased degree of 0
.
8and0
.
8, respec-
tively. We observe that
E
F
=
B
. According to the P-theory, by (15) we
correctly get
P
E∩F
(1
.
7m) =
P
B
(1
.
7m) = 0
.
7. However, the
Z
−
-system incor-
rectly gives
µ
E∩F
(1
.
7m) = min
∩
{
0
.
8
,
0
.
8
}
=0
.
8, while the B-system incorrectly
gives
µ
E∩F
(1
.
7m) = max
{
0
.
8+0
.
8
−
1
,
0
}
=0
.
6.
3.3 A Theorem: Where the Z-System and the B-System Work
or Fail
Theorem 2.
For A
⊆
U and B
⊆
U with P
A∩B
(
a
)
=0
, P
A∪B
(
a
)
=1
,
P
B|A
(
a
)
=1
and P
A|B
(
a
)
=1
, we have
(
i
)min
{
P
A
(
a
)+
P
B
(
a
)
,
1
}
>P
A∪B
(
a
)
>
max
{
P
A
(
a
)
,P
B
(
a
)
}
,
(
ii
)max
{
P
A
(
a
)+
P
B
(
a
)
−
1
,
0
}
<P
A∩B
(
a
)
<
min
{
P
A
(
a
)
,P
B
(
a
)
}
.
(19)
Proof.
(i) With
P
A∩B
(
a
)
= 0 or equivalently
P
A∩B
(
a
)
>
0, it follows from
(15)(a) that
P
A
(
a
)+
P
B
(
a
)
>P
A∪B
(
a
). Together with
P
A∪B
(
a
)
=1
or
P
A∪B
(
a
)
<
1, we get min
>P
A∪B
(
a
). More-
over, it follows from (15)(a) and (15)(b) that
P
A∪B
(
a
)=
P
B
(
a
)+
P
A
(
a
)
{
P
A
(
a
)+
P
B
(
a
)
,
1
}
−
P
A
(
a
)
P
B|A
(
a
)=
P
B
(
a
)+
P
A
(
a
)[1
−
P
B|A
(
a
)]
>P
B
(
a
), when
P
B|A
(
a
)
= 1. Similarly we also get
P
A∪B
(
a
)
>P
A
(
a
). That is,
P
A∪B
(
a
)
>
.
(ii) It follows from
P
A∪B
(
a
)
<
1 and (15)(a) that
P
A∪B
(
a
)=
P
A
(
a
)+
P
B
(
a
)
max
{
P
A
(
a
)
,P
B
(
a
)
}
−
P
A∩B
(
a
)
<
1or
P
A∩B
(
a
)
>P
A
(
a
)+
P
B
(
a
)
−
1, that is, max
{
P
A
(
a
)+
P
B
(
a
)
<P
A∩B
(
a
). Moreover, it follows directly from (15)(b) that
P
A∩B
(
a
)
<P
A
(
a
)when
P
B|A
(
a
)
<
1. Similarly we also get
P
A∩B
(
a
)
<
P
B
(
a
). That is,
P
A∩B
(
a
)
<
min
−
1
,
0
}
{
P
A
(
a
)
,P
B
(
a
)
}
.
=0or
P
A∩B
(
a
)
>
0 means that an over-
lap between
A
and
B
is measurable, while the condition
P
A∪B
(
a
)
=1or
P
A∪B
(
a
)
<
1 means that the difference between
U
and
A ∪ B
is measurable.
In the special case where
P
A∩B
(
a
)=0and
P
A∪B
(
a
) = 1, the B-system is
equivalent to the P-theory. This equivalence happens for the
The condition
P
A∩B
(
a
)
∪
operation only
if
P
A∪B
(
a
) = 1, and for the
operation only if
P
A∩B
(
a
)=0.
Moreover, when either
P
B|A
(
a
)=1or
P
A|B
(
a
) = 1, the B-system is
equivalent to the P-theory, as stated in Theorem 1.
To gain further insight, we rewrite (19) as
∩
B
A∪B
(
a
)
>P
A∪B
(
a
)
>Z
A∪B
(
a
)
≥
Z
A∩B
(
a
)
>P
A∩B
(
a
)
>
B
A∩B
(
a
)
,
(20)
B
A∪B
(
a
)=min
{
P
A
(
a
)+
P
B
(
a
)
,
1
}
,Z
A∪B
(
a
)=max
{
P
A
(
a
)
,P
B
(
a
)
}
,
Z
A∩B
(
a
)=min
{
P
A
(
a
)
,P
B
(
a
)
}
,
B
A∩B
(
a
)=max
{
P
A
(
a
)+
P
B
(
a
)
−
1
,
0
}
.
